Second year pure notes of a past AQA A Level Further Maths student, were incredibly useful to facilitate getting an A* in AQA Further Maths in 2024. Includes all second year pure topics of AQA further maths, cohesive and includes both notes and example questions. Studying mathematics at university ...
Zu W =
the solutions of zn = w
write modulus argument form
w in form De moivre's theorem :
includes fractions
a regular polygon
Use de moivres e zn un(cosno + isinno negative
with vertices on circle
isino)"
=
a
zn =
(r(coso + = r(cosno + isinno) powers
compare moduli & arguments centred at the origin
Write n different solutions :
stil
THINK ABOUT (4i) =
4
ARGAND FOR START VALUE
- arg( +
i) = arg(i) =
/
z 45 4i arg(45 + 4i) F
=
= +
Iti(cos + isin)
,
=
r (10530 + isin30) +
/6 ,
-
2π 30 =
(l i) 32(05 isin)
+ = +
828
*
145 + 4il = =
2
= 32(205 /2
+
+ isin/2)
= 32 ;
Roots of mity ze
Euler's formula :
nth roots of unity - 1 ,
ee ... cio-COSO + isino
reio-r(coso + isino
form a
regular n-gon on an argand diagram reio z= reio
ei - z
2
=
=
nth
root
S
=
Wi =
(e)" =
Wi = 0 1 2 ,3
. .
... n -
1
further factorising Z-zez we zei e2 + 3i = 22g
It Wi 0
each ROU is (2 +... + Wn e ((053 + isin3)
+ =
(2) (e)
a "
(z W((z-W*) [Re(W) + IWR
=
power of W
,
-
= z zws
wa
-
-
... + Wh
=
It Wit Wat ...
= I + 2 + + Z
= - 7 32
.
+ 1 . 04i
= 2e
Z" -81 z in Cartesian
= -w =
+ 3i
(e(n3)2
-
Sle iπr4 8)r 3
+3i
1- w
rugino 8) = =
32
e
= -
= =
W 4
: ezn3 el3insli
=
48 = , 3 5
0 =
:
Fifth ROU
, ,
4(c0s[ + isin)
isin (In 27)
* :
WY = W
9((0S(In27) +
13 3 i
=
* =
z +
wi (w2) =
e e, e
=
=
-8 89-1 38i
I 25 - 21
. .
Re(w") Re(w) = ,
, =
=
Re(w3) Re(w) =
product of two real factors
isinos
COS (z zi)(z zu) z2 352z + 9
Geometry
cosRe(W) = -
-
=
Re(W)
-
Sin50 + 5 sin40 + 10 sin30 + 10sin20 + 5 sinO
= 32(OS9 Jin
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